Homology Theory is a branch of mathematics that studies topological spaces through algebraic methods. It assigns a sequence of abelian groups or modules to a topological space, capturing its shape and structure. These groups, known as homology groups, help classify spaces based on their features, such as holes of different dimensions. The theory is fundamental in areas like algebraic topology and has applications in various fields, including data analysis and robotics. By comparing homology groups, mathematicians can determine when two spaces are equivalent, providing insights into their underlying properties and relationships.